rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Selected Works of Donald L. Burkholder
  name:
    name part: Davis, Burgess.
    role:
      role term: editor.
  name:
    name part: Song, Renming.
    role:
      role term: editor.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: xxu
    date issued: 2011
    edition: 1.
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XXV, 729 p. online resource.
  abstract: This book chronicles Donald Burkholder's thirty-five year study of martingales and its consequences. Here are some of the highlights. Pioneering work by Burkholder and Donald Austin on the discrete time martingale square function led to Burkholder and Richard Gundy's proof of inequalities comparing the quadratic variations and maximal functions of continuous martingales, inequalities which are now indispensable tools for stochastic analysis. Part of their proof showed how novel distributional inequalities between the maximal function and quadratic variation lead to inequalities for certain integrals of functions of these operators. The argument used in their proof applies widely and is now called the Burkholder-Gundy good lambda method. This uncomplicated and yet extremely elegant technique, which does not involve randomness, has become important in many parts of mathematics. The continuous martingale inequalities were then used by Burkholder, Gundy, and Silverstein to prove the converse of an old and celebrated theorem of Hardy and Littlewood. This paper transformed the theory of Hardy spaces of analytic functions in the unit disc and extended and completed classical results of Marcinkiewicz concerning norms of conjugate functions and Hilbert transforms. While some connections between probability and analytic and harmonic functions had previously been known, this single paper persuaded many analysts to learn probability. These papers together with Burkholder's study of martingale transforms led to major advances in Banach spaces. A simple geometric condition given by Burkholder was shown by Burkholder, Terry McConnell, and Jean Bourgain to characterize those Banach spaces for which the analog of the Hilbert transform retains important properties of the classical Hilbert transform. Techniques involved in Burkholder's usually successful pursuit of best constants in martingale inequalities have become central to extensive recent research into two well- known open prob…
  table of contents: On a class of stochastic approximation processes -- Sufficiency in the undominated case -- Iterates of conditional expectation operators -- On the order structure of the set of sufficient subfields -- Semi-Gaussian subspaces -- Successive conditional expectations of an integrable function -- Maximal inequalities as necessary conditions for almost everywhere convergence -- Martingale transforms. Ann. Math. Statist -- Extrapolation and interpolation of quasilinear operators on martingales -- A maximal function characterization of the class Hp -- Distribution function inequalities for the area integral -- Distribution function inequalities for martingales -- Boundary behaviour of harmonic functions in a half-space and Brownian motion -- One-sided maximal functions and Hp. J. Functional Analysis -- Boundary value estimation of the range of an analytic function -- A sharp inequality for martingale transforms -- Weak inequalities for exit times and analytic functions -- A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional -- Martingale transforms and the geometry of Banach spaces -- A nonlinear partial differential equation and the unconditional constant of the Haar system in Lp -- Boundary value problems and sharp inequalities for martingale transforms -- An elementary proof of an inequality of R. E. A. C. Paley -- Martingales and Fourier analysis in Banach spaces -- A sharp and strict Lp-inequality for stochastic integrals -- A proof of Pełczyn´ski’s conjecture for the Haar system -- Differential subordination of harmonic functions and martingales -- Explorations in martingale theory and its applications -- Strong differential subordination and stochastic integration -- The best constant in the Davis inequality for the expectation of the martingale square function -- Joseph L. Doob.
  note: edited by Burgess Davis, Renming Song.
  subject:
    topic: Statistics
  subject:
    topic: Statistics
  subject:
    topic: Statistical Theory and Methods
  classification: QA276-280
  classification: 519.5
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: Selected Works in Probability and Statistics
  identifier: 9781441972453
  identifier: http://dx.doi.org/10.1007/978-1-4419-7245-3
  record info:
    record creation date: 110217
    record change date: 20151125201751.0
    record identifier: 978-1-4419-7245-3